18.26.6 Gauduchon polar trace comparisons6C8B6C6B86C3

Proof. The form $\pi ^*\omega $ is semipositive, and it is positive on the nonempty open set where $\pi $ is an isomorphism, so $A_\pi >0$. By proposition 5.43.18, $u\circ \pi \in \mathrm{PSH}(Y_\pi ,\pi ^*\omega )$. Since $\mathrm{dd}^{\mathrm{c}}(\eta _\pi ^{n-1})=0$, distributional Stokes gives

Thus $\mu _\pi (u)$ is a positive measure of mass at most $A_\pi $, and the normalized total variation distance of two such measures is at most two. Symmetry and the triangle inequality persist under the supremum, which proves the pseudometric assertion.

Let $p,q$ be ordinary models with analytic singularities, choose a common log resolution $\pi $ in $\mathscr R$, and suppose that $\delta _\omega ^{\mathrm{pol}}(p,q)=0$. Then their measures in this coordinate agree. Let $E_ i$ be the prime components of the resolved ideals on $Y_\pi $, and $a_ i,b_ i$ the coefficients of $p$ and $q$ along them. The divisorial restriction argument of lemma 5.38.1 gives

Indeed, a smooth local correction makes the complex Hessian closed and positive without changing its coefficient mass on a divisor. Distinct components do not share trace mass on their intersections, and each component has strictly positive trace mass. Equality of these measures therefore implies $a_ i=b_ i$ for every $i$. The criterion of lemma 5.38.11, which uses only the finitely many divisors $E_ i$, applied in both directions gives $p\sim q$. An ordinary model is the ordinary envelope of every potential of its type, so ordinary models with the same type are equal. This proves separation.

On a curve, every smooth modification is an isomorphism and $\eta _\pi ^{n-1}=1$. Hence $A_\pi =\int _ X\omega $, and pullback preserves the total variation of polar curvature. The formula on curves and the assertions following it are therefore consequences of corollary 13.28.7, proposition 13.28.11.

Finally, $u-f$ has, in background $\omega '$, the same pulled-back curvature current and the same pole set as $u$ in background $\omega $, and Stokes' theorem gives $A_{\pi ,\omega '}=A_{\pi ,\omega }$. The gauge formula follows. It holds for the potentials themselves, before any ordinary projection. $\square$

Proof. The metric is finite, bounded by two, and separates points. A Cauchy sequence is Cauchy in total variation in each coordinate, and completeness of the space of finite signed measures gives a limit there; positivity and the mass bound pass to the limit. Given $\varepsilon >0$, choose $N$ such that every normalized coordinate distance is at most $\varepsilon $ for all $j,k\geq N$. Letting $k$ tend to infinity separately in each coordinate preserves this inequality, and taking the supremum proves convergence for $\widehat\delta $. The last assertion follows from the definition and the mass bound proved above. $\square$

Proof. Boundedly equivalent potentials have boundedly equivalent pullbacks on every modification, with the same pole set. Proposition 5.46.3 therefore gives equality of their polar currents and of their measures $\mu _\pi $ in every coordinate, which proves descent to ordinary classes.

If $p=P_\omega [p]$, the version of proposition 18.5.1 for ordinary envelopes gives

Thus $G_ f(p)\sim p-f$. Since ordinary envelopes depend only on the ordinary type, $P_{\omega '}[G_ f(p)]=P_{\omega '}[p-f]=G_ f(p)$, and similarly $P_\omega [G_ f(p)+f]=P_\omega [p]=p$. The same argument applied to models in background $\omega '$ proves surjectivity and the formula for the inverse. Invariance of the polar coordinates and the gauge identity of proposition 18.26.8 now give

Subtraction of a smooth function and bounded equivalence preserve analytic singularities. If $p=P_\omega [p]_{\mathcal I}$, the gauge estimate for ideal envelopes gives $P_{\omega '}[p-f]_{\mathcal I}\sim p-f\sim G_ f(p)$. By corollary 18.4.4, this ideal envelope is an ordinary model, and two ordinary models of the same ordinary type coincide. Hence it equals $G_ f(p)$, which shows that $G_ f$ maps ideal models to ideal models; the inverse map gives the converse. $\square$

Proof. By proposition 5.43.18, the pullbacks belong to $\mathrm{PSH}(Y_\pi ,\pi ^*\omega )$, and proposition 5.46.7, applied with background $\pi ^*\omega $, shows that their polar currents are closed and positive. Both currents are carried by the union of the two Borel pole sets, which has Lebesgue measure zero. Equality of their $\eta _\pi $-traces therefore gives equality of the currents by lemma 5.46.4, and the converse is immediate. Since every $A_\pi $ is positive, the last assertion follows. $\square$

Proof. Fix $\pi $. By proposition 18.26.9, the measures $\mu _\pi (p_ j)$ converge in total variation to a positive measure $\lambda _\pi $. Put

This is a Borel set of Lebesgue measure zero. Every measure $\mu _\pi (p_ j)$ vanishes on its complement, so $\lambda _\pi (Y_\pi \setminus B_\pi )=0$. The polar currents are closed and positive by corollary 18.26.11, and their $\eta _\pi $-trace masses are at most $A_\pi $. By compactness of positive currents of bounded mass, a subsequence converges weakly. Its limit $R$ is closed and positive, and $R\wedge \eta _\pi ^{n-1}=\lambda _\pi $. Since positivity bounds all coefficient measures by the Hermitian trace, $R$ is carried by $B_\pi $. Any two subsequential limits have the same null carrier and the same trace, so they agree by lemma 5.46.4. As every subsequence has a further convergent subsequence, the whole sequence of currents converges weakly to this unique limit $R_\pi $.

Apply proposition 5.46.5 to $g$, with background $\pi ^*\omega $ and potential $p_ j\circ \pi $. It gives $g_*\operatorname {Pol}_{\pi '}(p_ j)=\operatorname {Pol}_\pi (p_ j)$. Proper pushforward is weakly continuous, since a smooth test form downstairs pulls back to a smooth test form upstairs, and passing to the limit proves compatibility. The limits are constructed separately in each coordinate, so no diagonal subsequence over $\mathscr R$ is needed.

To obtain one pole carrier, put $h_ m=\sum _{j=1}^ m2^{-j}p_ j$. These functions decrease, and they are $\omega $-plurisubharmonic because the coefficients have sum at most one and $\omega >0$. Normalized compactness, theorem 5.25.2, bounds their $L^1$ norms uniformly, so corollary 5.25.3 gives a limit $h\in \mathrm{PSH}(X,\omega )$. Since all summands are nonpositive, each pole set of a $p_ j$ is contained in the pole set of $h$. Hence $B_\pi \subset \{ h\circ \pi =-\infty \} $, which gives the last assertion. $\square$

Proof. At $x=[1:0:0]$, put

The homogeneous quotient defines a global nonpositive potential. Its curvature is positive, since $\log (|Z_1|^2+|Z_2|^2)$ is plurisubharmonic in a local trivialization and the remaining background is $(1-a)\omega $. Its only pole is $x$, where it is $a\log |z|^2+O(1)$. By proposition 18.4.8, $p_ x=P_\omega [u_ x]$ is an ideal model with analytic singularities, ordinarily equivalent to $u_ x$. Unitary projective automorphisms preserve $\omega $ and commute with its envelopes for the zero obstacle. Transporting this construction to pairwise distinct points $x_ j$, we obtain models $p_ j$.

Fix $\pi :Y_\pi \to X$ and some Gauduchon form $\gamma _\pi $ on $Y_\pi $. The closed polar current of $p_ j\circ \pi $ is supported on the analytic fiber over $x_ j$. The divisorial restriction in lemma 5.38.1 and the support theorem, [Chapter III, Corollaries 2.11 and 2.14, pp. 141–143, DemCADG], give

where the sum runs over the irreducible curves $E\subset Y_\pi $ with $\pi (E)=x_ j$. Indeed, corollary 18.26.11 shows that the polar current is closed, and after the curve components are removed, the remaining closed current of order zero is supported at finitely many points and therefore vanishes. The sum is finite. Every such curve is contracted by $\pi $, so the restriction of $\pi ^*\omega $ to it is zero. Put

A modification of a surface has finitely many exceptional curves, so there are only finitely many points over which $\pi $ has exceptional curves. Since the points $x_ j$ are distinct, $J_\pi $ is finite. Choose $\varepsilon _\pi >0$ with $\varepsilon _\pi C_{\pi ,j}\leq 1/j$ for every $j\in J_\pi $; if $J_\pi $ is empty, take $\varepsilon _\pi =1$. Now fix

This form is positive and Gauduchon, since on a surface the Gauduchon equation is $\mathrm{dd}^{\mathrm{c}}\eta _\pi =0$. Moreover

These forms are fixed for the whole sequence, and the estimate holds for every modification. Since $\mu _\pi (0)=0$, taking the supremum gives the asserted convergence.

To compute the singularity volumes, let $\rho :Y\to X$ be the blowup at $x$, with exceptional curve $E$, and let $F:Y\to \mathbb P^1$ be the morphism resolving the pencil $[Z_1:Z_2]$. Write $\omega _1$ for the normalized Fubini–Study form on $\mathbb P^1$. Then

For every $v\sim u_ x$, divisorial removal gives $H_ a+\mathrm{dd}^{\mathrm{c}}w\geq 0$, where $w=v\circ \rho -u_ x\circ \rho $ is bounded after extension across $E$. The bounded product of proposition 5.28.2 assigns no mass to $E$, so change of variables off $E$ identifies the non-pluripolar mass of $v$ with $\int _ Y(H_ a+\mathrm{dd}^{\mathrm{c}}w)^2$. Since $H_ a$ is closed, integration by parts for bounded potentials gives

For the last equality, note that $H_ a$ represents the class $H-aE$, where $H=\rho ^*[\omega ]$, and that $H^2=1$, $H\cdot E=0$ and $E^2=-1$. Hence both singularity volumes of $p_ x$, and by unitary transport those of every $p_ j$, are equal to $1-a^2$. The zero model has both volumes equal to one. Since $p_ j\leq 0$, definition 13.22.1 gives $d_\omega (p_ j,0)=1-(1-a^2)=a^2$. This also proves the common positive lower bound for the volumes and the incompatibility of the topologies. $\square$